Transformations for denoising images

ABSTRACT

Systems and methods of denoising images are described. In one aspect, spatially-shifted forward transforms of the input image are computed. Each forward transform is computed based on a denoiser transform Z having an associated transpose Z′, wherein a matrix multiplication between Z and Z′ produces a diagonal matrix Λ, Z=F(D), F specifies a mapping from coefficients of D to coefficients of Z, and D substantially corresponds to a frequency-domain transform. The forward transforms are denoised based on nonlinear mappings derived from quantization values linked to the input image. Spatially-shifted inverse transforms of the denoised forward transforms are computed. Each inverse transform is computed based on Z and Z′. An output image is computed based on a combination of spatially-shifted inverse transforms.

TECHNICAL FIELD OF THE INVENTION

This invention relates to transformations for denoising images.

BACKGROUND OF THE INVENTION

Digital images and video frames are compressed in order to reduce datastorage and transmission requirements. In most image compressionmethods, certain image data is discarded selectively to reduce theamount of data needed to represent the image while avoiding substantialdegradation of the appearance of the image.

Transform coding is a common image compression method that involvesrepresenting an image by a set of transform coefficients. The transformcoefficients are quantized individually to reduce the amount of datathat is needed to represent the image. A representation of the originalimage is generated by applying an inverse transform to the transformcoefficients. Block transform is coding is a common type of transformcoding method. In a typical block transform coding process, an image isdivided into small rectangular regions (or “blocks”), which aresubjected to forward transform, quantization and coding operations. Manydifferent kinds of block transforms may be used to encode the blocks.Among the common types of block transforms are the cosine transform(which is the most common), the Fourier transform, the Hadamardtransform, and the Haar wavelet transform. These transforms produce anM×N array of transform coefficients from an M×N block of image data,where M and N have integer values of at least 1.

The quality of an image often is degraded by a block transform codingprocess. For example, discontinuities often are introduced at the blockboundaries in the reconstructed image and ringing artifacts often areintroduced near image boundaries.

Different approaches have been proposed for enhancing compressed imagesby reducing the appearance of artifacts introduced by block transformcoding processes. Among the most common image enhancement approaches areapproaches that filter along block boundaries, approaches that optimizeby projecting onto convex sets, and approaches that perform wavelettransform thresholding. Another approach for enhancing compressed imagesinvolves the reapplication of shifted JEPG transforms to a JPEG image.In particular, this approach re-applies the JPEG compression operator toshifted versions of the already-compressed image and averages all of theresulting images to produce an enhanced output image.

SUMMARY OF THE INVENTION

The invention features systems and methods of denoising images.

The invention features systems and methods in which spatially-shiftedforward transforms of the input image are computed. Each forwardtransform is computed based on a denoiser transform Z having anassociated transpose Z′, wherein a matrix multiplication between Z andZ′ produces a diagonal matrix Λ, Z=F(D), F specifies a mapping fromcoefficients of D to coefficients of Z, and D substantially correspondsto a frequency-domain transform. The forward transforms are denoisedbased on nonlinear mappings derived from quantization values linked tothe input image. Spatially-shifted inverse transforms of the denoisedforward transforms are computed. Each inverse transform is computedbased on Z and Z′. An output image is computed based on a combination ofspatially-shifted inverse transforms.

Other features and advantages of the invention will become apparent fromthe following description, including the drawings and the claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of a system for compressing an original image.

FIG. 2 is a flow diagram of a block-transform-based method ofcompressing an original image.

FIG. 3 is a flow diagram of an embodiment of a method of processing acompressed image to produce an output image characterized by reducedcompression artifacts.

FIG. 4 is a block diagram of an embodiment of an image processing systemfor implementing output image generator steps in the method of FIG. 3.

FIG. 5 is a graph of threshold output in an implementation of thenonlinear denoiser module of FIG. 4 plotted as a function of inputtransform coefficient values.

FIG. 6 is a block diagram of an implementation of the output imagegenerator module in of FIG. 4.

FIG. 7 is a diagrammatic view of a set of intermediate images and aringing correction image generated from the set of intermediate images.

FIG. 8 is a block diagram of a pair of concentric windows used tocompute a measure of block contrast and a measure of local contrast frompixel values of a base image.

FIG. 9 are graphs of weighting values plotted as functions of contrastdifference between measures of block contrast and local contrastcomputed for a pixel of a base image.

DETAILED DESCRIPTION OF THE INVENTION

In the following description, like reference numbers are used toidentify like elements. Furthermore, the drawings are intended toillustrate major features of exemplary embodiments in a diagrammaticmanner. The drawings are not intended to depict every feature of actualembodiments nor relative dimensions of the depicted elements, and arenot drawn to scale.

I. Compressing an Original Image

FIG. 1 shows a prior art method of processing an original image 10 toproduce a compressed image 12. In accordance with the illustratedmethod, a forward transform 14 is applied to the original image 10 toproduce a set of forward transform coefficients 16. The forwardtransform 14 may be any type of transform that is operable to convertthe original image 10 into a set of coefficients in a selected domain. Aquantizer 18 is applied individually to the forward transformcoefficients 16 to produce a set of quantized forward coefficients 20.The quantizer 18 discards some of forward transform coefficientinformation, enabling the original image 10 to be compressed. An encoder22 encodes the quantized forward transform coefficients using any typeof lossless encoding technique to produce the compressed image 12.

Original image 10 may be a binary image (e.g., a dark and bright dotpattern), a multilevel single-color image (e.g., a gray-level image), ora multilevel multi-color image. In general, the image compressionprocess of FIG. 1 is applied individually to each color plane of theoriginal image 10.

FIG. 2 shows a prior art block-transform-based image method ofcompressing original image 10. In accordance with this method, originalimage 10 initially is converted into a preselected luminance-based colorspace (e.g., the YCrCb color space), if the original image 10 is notalready specified in the preselected color space (block 24). Each colorplane of the image in the preselected color space corresponds to arespective image (i.e., an array of pixel values) that is processedindividually as follows. The color components (e.g., the Cr and Cb colorcomponents) are downsampled (block 26). Each color plane is divided intoblocks of pixels (e.g., 8×8 pixel blocks) (block 28). A block transformis applied to each pixel block individually (block 30). Any kind ofblock transform may be applied to the blocks. Exemplary types of blocktransforms include the cosine transform, Fourier transform, Hadamardtransform, and Haar wavelet transform. The resulting transformcoefficients are quantized (block 32). The quantized transformcoefficients are encoded using a lossless coding technique to producecompressed image 12 (block 34).

II. Reducing Artifacts in Compressed Images

The embodiments described below are configured to reduce artifactsinherently introduced by processes that are used to create compressedimages. In many instances, these embodiments reduce image compressionartifacts without degrading image quality, such as by blurring featuresin the image. As described in detail below, some implementations ofthese embodiments are particularly well-suited to substantially reduceringing and blocking compression artifacts that are introduced byblock-transform-based image compression techniques, such as blockdiscrete cosine transform (DCT) image compression techniques.

FIG. 3 shows an embodiment of a method of processing a compressed image12 that is generated by the block-transform-based image compressionmethod of FIG. 2 to produce an output image 40 with reduced compressionartifacts. In this method, each plane of the compressed image 12 isprocessed individually. The compressed image data initially isdecompressed (block 42). Spatially-shifted forward transforms arecomputed from the decompressed image data 62 (block 44). In thisprocess, a forward transform operation is applied to each of multipleshifted versions of the decompressed image data 62 to produce multiplerespective sets of forward transform coefficients 46. For example, in animplementation in which the image 12 is compressed based on blocks ofM×N pixels, the forward transform operation is applied to thedecompressed image data 62 on a subset containing K shifts from the M×Nindependent shifts possible in an M×N transform to produce K sets offorward transform coefficients, where K, M, and N have integer values ofat least 1. In one exemplary implementation, both M and N have a valueof 8.

The forward transform coefficients 46 of each set are nonlinearlytransformed (block 48). An inverse transform operation (block 50) isapplied to the sets of nonlinearly transformed forward transformcoefficients 46 to produce respective intermediate images. As explainedin detail below, the intermediate images are combined to reducecompression artifacts in each color plane 64 (block 52). The colorcomponent image planes (e.g., Cr and Cb) are upsampled to the originalresolution and the resulting image planes are converted back to thecolor space (e.g., the Red-Green-Blue color space) of the original image10 (block 54). The image planes are combined to produce the output image40.

FIG. 4 shows an embodiment of a system 60 for processing thedecompressed image data 62 generated by the method of FIG. 3 to producethe compression-artifact-reduced output image 40. Processing system 60includes a forward transform module 66, a nonlinear denoiser module 68,an inverse transform module 70, and an output image generator module 72.In general, the modules 66-72 of system 60 are not limited to anyparticular hardware or software configuration, but rather they may beimplemented in any computing or processing environment, including indigital electronic circuitry or in computer hardware, firmware, devicedriver, or software. For example, in some implementations, these modules66-72 may be embedded in the hardware of any one of a wide variety ofelectronic devices, including digital cameras, printers, and portableelectronic devices (e.g., mobile phones and personal digitalassistants). In addition, although full (or complete) input images areprocessed in the illustrated embodiments, other embodiments may beconfigured to sequentially process a series of sub-image portions (e.g.,swaths) of an input image.

A. Forward Transform Module

The forward transform module 66 computes from the decoded image data 62K sets (C₁, C₂, . . . , C_(K)) of shifted forward transforms,corresponding to K unique positions of a blocking grid relative to thecompressed image 12. As explained in detail below, a class of forwardtransforms is used that enables artifacts in the decompressed image 62to be reduced substantially in substantially multiplier-free hardwareimplementations.

In general, each block of the forward transform is computed based on adenoiser transform Z having an associated transpose Z′, wherein a matrixmultiplication between Z and Z′ produces a diagonal matrix Λ, Z=F(D), Fspecifies a mapping from coefficients of D to coefficients of Z, and Dsubstantially corresponds to a frequency-domain transform. Each block ofthe spatially-shifted forward transforms C₁ (i=1, 2, . . . , K) may becomputed based on the separable application of the denoiser transform Zin two dimensions as follows:B=ZXZ′  (1)where X corresponds to a block of the decompressed image data 62, and Bcorresponds to the transform coefficients of the image block X. In someimplementations, the mapping function F specifies a mapping fromcoefficients of D to coefficients of Z having values selected from 0 and±2^(N) where N has an integer value. This allows the matrixmultiplication specified in equation (1) to be computed trivially oreasily by simple binary shifts in software or may be implemented inhardware without requiring a multiplier.

The mapping function F may specify a mapping from weighted coefficientsof D to corresponding coefficients of Z. In these implementations, thecoefficients of D are weighted by respective scaling factors, which maybe the same or different. In one exemplary embodiment, the mappingfunction F corresponds to a rounding operator applied to the weightedcoefficients of D. That is:Z _(ij)=round{α_(ij) ·D _(ij)}  (2)wherein round{ } is the rounding operator, Z_(ij) is the coefficient ofZ in row i and column j, D_(ij) is the coefficient of D in row i andcolumn j, and α_(ij) is the respective scaling factor for D_(ij). In oneimplementation, the coefficients of D are weighted by a common (i.e.,the same) scaling factor (i.e., α_(ij)=α for all i,j), in which case:Z _(ij)=round{α·D _(ij)}  (3)

Other exemplary mapping functions include arithmetic operators, such asthe floor operator, the ceiling operator, and the truncate operator. Thefloor operator returns the largest integer less than or equal to theargument. The ceiling operator returns the smallest integer greater thanor equal to the argument. The truncate operator functions like the flooroperator for positive argument values and like the ceiling operator fornegative argument values. Still other mapping functions that produceinteger approximations to the coefficients of D may be used to generatethe coefficients of the denoiser transform Z.

In some implementations, D is a block-based linear transform, such as adiscrete cosine transform (DCT). In one dimension, the DCT transform isgiven to four decimal places by the following 8 by 8 matrix:

$\begin{matrix}{D = \begin{matrix}0.3536 & 0.3536 & 0.3536 & 0.3536 & 0.3536 & 0.3536 & 0.3536 & 0.3536 \\0.4904 & 0.4157 & 0.2778 & 0.0975 & {- 0.0975} & {- 0.2778} & {- 0.4157} & {- 0.4904} \\0.4619 & 0.1913 & {- 0.1913} & {- 0.4619} & {- 0.4619} & {- 0.1913} & 0.1913 & 0.4619 \\0.4157 & {- 0.0975} & {- 0.4904} & {- 0.2778} & 0.2778 & 0.4904 & 0.0975 & {- 0.4157} \\0.3536 & {- 0.3536} & {- 0.3536} & 0.3536 & 0.3536 & {- 0.3536} & {- 0.3536} & 0.3536 \\0.2778 & {- 0.4904} & 0.0975 & 0.4157 & {- 0.4157} & {- 0.0975} & 0.4904 & {- 0.2778} \\0.1913 & {- 0.4619} & 0.4619 & {- 0.1913} & {- 0.1913} & 0.4619 & {- 0.4619} & 0.1913 \\0.0975 & {- 0.2778} & 0.4157 & {- 0.4904} & 0.4904 & {- 0.4157} & 0.2778 & {- 0.0975}\end{matrix}} & (4)\end{matrix}$

Applying equation (3) with α=3.5 to the one-dimensional DCT transform ofequation (4) yields the following denoiser transform:

$\begin{matrix}{Z = \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\2 & 1 & 1 & 0 & 0 & {- 1} & {- 1} & {- 2} \\2 & 1 & {- 1} & {- 2} & {- 2} & {- 1} & 1 & 2 \\1 & 0 & {- 2} & {- 1} & 1 & 2 & 0 & {- 1} \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 \\1 & {- 2} & 0 & 1 & {- 1} & 0 & 2 & {- 1} \\1 & {- 2} & 2 & {- 1} & {- 1} & 2 & {- 2} & 1 \\0 & {- 1} & 1 & {- 2} & 2 & {- 1} & 1 & 0.\end{matrix}} & (5)\end{matrix}$In this implementation, the diagonal matrix Λ satisfying the conditionthat ZZ′=Λ is given by equation (5):

$\begin{matrix}{A = \begin{matrix}8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 12 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 20 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 12 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 12 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 20 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 12.\end{matrix}} & (6)\end{matrix}$Other exemplary values for the scaling factor α in equation (3) include3.0 and 2.5.

In some implementations, the blocks of the spatially-shifted forwardtransforms (C₁, C₂, . . . , C_(K)) are computed based on a factorizationof the denoiser transform Z. For example, the denoiser transform ofequation (5) may be implemented by a fast factorization Z=PR₃R₂R₁, whereP is a permutation matrix, and R₁, R₂, And R₃ are sparse factor matricesgiven by:

$\begin{matrix}{P = \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\end{matrix}} & (7) \\{R_{3} = \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\1 & {- 1} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 2 & 0 & 0 & 0 & 0 \\0 & 0 & {- 2} & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & {- 2} & 1 \\0 & 0 & 0 & 0 & 0 & 1 & 1 & 2 \\0 & 0 & 0 & 0 & {- 2} & 1 & {- 1} & 0 \\0 & 0 & 0 & 0 & {- 1} & {- 2} & 0 & 1\end{matrix}} & (8) \\{R_{2} = \begin{matrix}1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & {- 1} & 0 & 0 & 0 & 0 & 0 \\1 & 0 & 0 & {- 1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{matrix}} & (9) \\{R_{1} = \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & {- 1} & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & {- 1} & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & {- 1} & 0 \\1 & 0 & 0 & 0 & 0 & 0 & 0 & {- 1}\end{matrix}} & (10)\end{matrix}$

Computer code for an implementation of the fast factorization shown inequations (7)-(10) for the forward denoiser transform of equation (5)and the corresponding factorization for the corresponding inversedenoiser transform (Z⁻¹) is contained in the attached Appendix.

In some other implementations, D is a wavelet-based decompositiontransform. In one of these implementations, for example, D may be aforward discrete wavelet transform (DWT) that decomposes aone-dimensional (1-D) sequence (e.g., line of an image) into twosequences (called sub-bands), each with half the number of samples. Inthis implementation, the 1-D sequence may be decomposed according to thefollowing procedure: the 1-D sequence is separately low-pass andhigh-pass filtered by an analysis filter bank; and the filtered signalsare downsampled by a factor of two to form the low-pass and high-passsub-bands.

B. Nonlinear Denoiser Module

The nonlinear denoiser module 68 nonlinearly transforms the outputtransform coefficients that are computed from the sets (C₁, C₂, . . . ,C_(K)) of shifted forward transforms that are computed by the forwardtransform module 66.

In some implementations, the sets of forward transform coefficients aretransformed in accordance with respective nonlinear thresholdingtransformations (T_(1,) T₂, . . . , T_(K)). FIG. 5 shows a graph 74 ofthreshold output plotted as a function of input coefficient values inaccordance with an embodiment of the denoiser module 68. In thisembodiment, the forward transform coefficients are nonlinearlytransformed by setting to zero each coefficient with an absolute valuebelow a respective threshold (t_(ij), where i,j refer to the indices ofthe quantization element, with i having values in the range of 0 to M-1and j having values in the range of 0 to N-1) and leaving unchanged eachcoefficient with an absolute value equal to or above a respectivethreshold (t_(ij)). Quantization matrices 76 (or “Q Matrices”) are usedto set the parameters t_(ij) for the nonlinear thresholdingtransformations (T₁, T₂, . . . , T_(K)). In some implementations, thequantization matrices contain the same quantization parameters q_(ij)that were originally used to compress image 12. These quantizationparameters may be stored in the compressed image 12 in accordance with astandard image compression scheme (e.g., JPEG).

In some implementations, the threshold parameters t_(ij) are set inblock 77 by a function M that maps the quantization parameters q_(ij) ofthe Q matrices to the corresponding threshold parameters t_(ij). In someof these implementations, the function M is an affine function that mapsthe quantization parameters q_(ij) to the t_(ij) in accordance withequation (11):t _(ij)=β_(ij) ·q _(ij)+γ_(ij)  (11)where β_(ij) and γ_(ij) are respective constants. The β_(ij) may be thesame or different. Similarly, the γ_(ij) may be the same or different.In some embodiments implemented in accordance with the forward denoisertransform implementations specified in equation (3), γ_(ij)=0 andβ_(ij)=α_(ij) for all i,j (i.e., the scaling factors used to compute thecoefficients of Z from the coefficients of D are the same as the scalingfactors used to compute the thresholds t_(ij)).

Based on equation (1) and the diagonal matrix condition ZZ′=Λ, one mayderive the identity matrix condition Z′Λ⁻¹Z=I. The block X of thedecompressed image data 62 may be recovered as follows:X=Z′[Λ⁻¹BΛ⁻¹]Z  (12)where the quantity in brackets of equation (12) determines the scalingof the coefficients B that needs to be applied before multiplying by Z′on the left and multiplying by Z on the right. Since the quantity inbrackets involves multiplying by a diagonal matrix on the right and adiagonal matrix on the left, the scaling factor for the i,j element of Bis given by component-wise multiplication by a matrix S with componentsgiven by s_(ij)=λ_(i) ⁻¹λ_(j) ⁻¹, where λ_(i) ⁻¹ is the inverse of thei^(th) diagonal element of the matrix Λ. In the implementation ofequations (5) and (6), S is given to four decimal places by:

$\begin{matrix}{S = \begin{matrix}0.0156 & 0.0104 & 0.0063 & 0.0104 & 0.0156 & 0.0104 & 0.0063 & 0.0104 \\0.0104 & 0.0069 & 0.0042 & 0.0069 & 0.0104 & 0.0069 & 0.0042 & 0.0069 \\0.0063 & 0.0042 & 0.0025 & 0.0042 & 0.0063 & 0.0042 & 0.0025 & 0.0042 \\0.01047 & 0.0069 & 0.0042 & 0.0069 & 0.0104 & 0.0069 & 0.0042 & 0.0069 \\0.0156 & 0.0104 & 0.0063 & 0.0104 & 0.0156 & 0.0104 & 0.0063 & 0.0104 \\0.0104 & 0.0069 & 0.0042 & 0.0069 & 0.0104 & 0.0069 & 0.0042 & 0.0069 \\0.0063 & 0.0042 & 0.0025 & 0.0042 & 0.0063 & 0.0042 & 0.0025 & 0.0042 \\0.0104 & 0.0069 & 0.0042 & 0.0069 & 0.0104 & 0.0069 & 0.0042 & 0.0069\end{matrix}} & (13)\end{matrix}$

In some implementations, the non-trivial multiplications that arespecified in equations (12) and (13) are absorbed in the nonlineardenoising computations performed by module 68. In some of theseimplementations, the nonlinear denoiser module 68 modifies the forwardtransform coefficients C_(ij) in accordance with equation (14), whichincorporates the nonlinear transform of FIG. 5:

$\begin{matrix}\begin{matrix}{\Gamma_{ij} = {s_{ij} \cdot C_{ij}}} & {{{if}\mspace{14mu}{s_{ij} \cdot C_{ij}}} \geq t_{ij}} \\{= 0} & {{otherwise}.}\end{matrix} & (14)\end{matrix}$where Γ_(ij) are the denoised coefficients output from the nonlineardenoiser module 68.

In some embodiments, the nonlinear denoiser module 68 also incorporatessharpness enhancements by modifying the nonlinearities, T_(i). Inparticular, nonlinear denoiser module 68 multiplies the nonlineartransform parameter values q_(ij) by gain factors g_(ij). There are manyways to set the gain factors. In one implementation, the gain factorsare given by equation (15):g _(ij)=1.0+φ*(i+j)/14  (15)where i and j are the indices selecting each quantizer coefficient. Inimplementations designed for 8×8 blocking grids, the values of i and jare in the range [0-7], with 0 representing the lowest spatialfrequency, and 7 representing the highest spatial frequency. The valueof sharpness factor, s, determines the amount of sharpening. When φ=0,no sharpening is applied. Values of φ between 0.5 and 1.5 provide anadequate sharpness range to compensate for slight blurring that might becaused by the output image generator methods. A matrix containingexemplary gain factors computed from equation (1) for φ=0.5 is providedin Table 1, below:

TABLE 1 Gain Factors g_(ij) for φ = 0.5 1.0000 1.0357 1.0714 1.10711.1429 1.1786 1.2143 1.2500 1.0357 1.0714 1.1071 1.1429 1.1786 1.21431.2500 1.2857 1.0714 1.1071 1.1429 1.1786 1.2143 1.2500 1.2857 1.32141.1071 1.1429 1.1786 1.2143 1.2500 1.2857 1.3214 1.3571 1.1429 1.17861.2143 1.2500 1.2857 1.3214 1.3571 1.3929 1.1786 1.2143 1.2500 1.28571.3214 1.3571 1.3929 1.4286 1.2143 1.2500 1.2857 1.3214 1.3571 1.39291.4286 1.4643 1.2500 1.2857 1.3214 1.3571 1.3929 1.4286 1.4643 1.5000

C. Inverse Transform Module

The inverse transform module 70 computes sets (C⁻¹ ₁, C⁻¹ ₂, . . . , C⁻¹_(K)) of inverse transforms from the sets of nonlinearly transformedforward transform coefficients. The inverse transform module 70 appliesthe inverse of the forward transform operation that is applied byforward transform module 66. The outputs of the inverse transform module70 are intermediate images (I₁, I₂, . . . , I_(K)) representing theimage data in the spatial domain. The terms inverse transforms (C⁻¹ ₁,C⁻¹ ₂, . . . , C⁻¹ _(K)) and intermediate images (I₁, I₂, . . . , I_(K))are used synonymously herein. The blocks of the spatially-shiftedinverse transforms (C⁻¹ ₁, C⁻¹ ₂, . . . , C⁻¹ _(K)) may be computed byreplacing B in equation (12) with the output of the denoiser module 68.

D. Output Image Generator Module

The output image generator module 72 combines the intermediate images(I₁, I₂, . . . , I_(K)) to form the image planes 64 of the final outputimage 40. In general, the output image generator module 72 computesoutput image 40 based on a combination of intermediate images (I₁, I₂, .. . , I_(K)) For example, in some implementations, the output image 40is computed from a weighted combination of the intermediate images (I₁,I₂, . . . , I_(K)) In one of these implementations, the output image 40corresponds to an average of the intermediate images (I₁, I₂, . . . ,I_(K)).

FIG. 6 shows an embodiment of the output image generator module 72 thatincludes a base image generator 80 and a ringing correction imagegenerator 82.

The base image generator 80 computes a base image 84 corresponding to anestimate of the original uncompressed image from a combination of theintermediate images (I₁, I₂, . . . , I_(K)) In the illustratedembodiment, base image generator 80 computes a base image (I_(AVE)) thathas pixel values corresponding to averages of corresponding pixels inthe intermediate images (I₁, I₂, . . . , I_(K)).

The ringing correction image generator 82 computes a ringing correctionimage (I_(RC)) based on the intermediate images (I₁, I₂, . . . , I_(K))In some implementations, the ringing correction image generator 82computes a ringing correction image 86 (I_(RC)) based at least in parton measures of local spatial intensity variability that are computed forpixels of the intermediate images (I₁, I₂, . . . , I_(K)) For example,in a minimum variance de-ringing implementation, the value of a givenpixel of the ringing correction image 86 is set to the value of thecorresponding intermediate image pixel having the lowest computedmeasure of spatial intensity variability of all the intermediate imagepixels corresponding to the pixel in the ringing correction image 86. Ina weighted variance de-ringing approach, each pixel value of the ringingcorrection image 86 is assigned a value corresponding to an average ofmultiple corresponding intermediate image pixels in a lowest percentile(e.g., the 30% percentile) of local spatial variability measures of allthe intermediate image pixels corresponding to the given pixel in theringing correction image 86. The weighted variance de-ringing approachhas been observed to reduce ringing compression artifacts while avoidingnoise that sometimes is introduced by the minimum variance de-ringingapproach.

Referring to FIG. 7, in some embodiments, spatial intensity variabilityis measured by the spatial variance (σ₁ ², σ₂ ², . . . , σ_(K) ²), whichis computed for pixels of the intermediate images (I₁(x,y), I₂(x,y), . .. , I_(K)(x,y)) corresponding to a given pixel I_(RC)(x,y) in theringing correction image 86. The spatial variance measures are computedbased on respective intermediate image regions defined by respectivewindows 90, 92, 94, 96. In the illustrated embodiment, the windows 90-96have dimensions of 3 pixels in width by 3 pixels in height. In oneexemplary implementation, the spatial variance (σ₀) for a givenintermediate image pixel I_(j)(x,y) is given by equation (17):

$\begin{matrix}{\sigma_{0}^{2} = {\frac{1}{K} \cdot {\sum\limits_{j = 1}^{K}\left( {V_{j} - \left\langle V \right\rangle} \right)^{2}}}} & (16)\end{matrix}$where <V> is the average pixel value in the neighborhood of the givenpixel

$\left( {{i.e.},{\left\langle V \right\rangle = {\frac{1}{K}{\sum\limits_{i = 1}^{K}V_{i}}}}} \right)$and K is the number (e.g., nine) of pixels in the neighborhood of thegiven pixel in the ringing correction image 86. In the illustratedembodiment, the value of the intermediate image pixel (e.g.,(I_(MIN)(x,y)) with the smallest computed variance (σ_(MIN) ²) isassigned to the corresponding pixel (I_(RC)(x,y)) in the ringingcorrection image 86.

In another exemplary implementation, the spatial variability (Δ₀) for agiven intermediate image pixel I_(j)(x,y) with a value V₀ is given byequation (18):

$\begin{matrix}{\Delta_{0}^{2} = {\frac{1}{K} \cdot {\sum\limits_{j = 1}^{K}\left( {V_{j} - V_{0}} \right)^{2}}}} & (17)\end{matrix}$

It has been observed that the base image 84 is characterized by asubstantial reduction in the appearance of compression artifacts, suchas blocking artifacts, but in some cases the base image 84 stillcontains residual ringing artifacts. It also has been observed that theringing correction image 86 (I_(RC)) is characterized by substantiallyreduced ringing compression artifacts relative to the base image 84, butotherwise is generally more blurry than the base image 84. Based onthese observations, the output image generator module 72 is configuredto combine the base image 84 and the ringing correction image 86 in waysthat substantially reduce the appearance of compression artifacts in theresulting output image 40.

In the embodiment of FIG. 6, the output image generator module 72includes a weighted sum generator 98 that computes weighted sums ofcorresponding pixels of the base image 84 (I_(AVE)(x,y)) and the ringingcorrection image 86 (I_(RC)(x,y)) to generate the values of the pixelsof the output image 40 in accordance with equation (19):I _(OUT)(x,y)=α(x,y)·I _(RC)(x,y)+(1−α(x,y))·I _(AVE)(x,y)  (18)where α has a value in the range [0,1].

A weights generator module 100 computes the values of α(x,y) for eachpixel of the output image 40. In general, for pixels that aresufficiently far from sharp transitions, where the expected ringing isminimal, the weights generator module 100 sets α(x,y)≅0. For pixels on asharp transition, the weights generator module 100 also sets α(x,y)≅0 inorder to not blur the transition. For pixels that are sufficiently closeto (e.g., adjacent to) but not on a sharp transition, the weightsgenerator module 100 sets α(x,y)≅1 to reduce the potential ringing.

In some embodiments, the weights generator module 100 computes α(x,y)for a given pixel based on a measure of relative contrast between alocal region in a spatial window encompassing the given pixel and alarger region in a spatial window surrounding the local region. In someimplementations, image contrast is computed from the pixel range, whichis the difference, in a spatial window, between maximum and minimumpixel values. As shown in FIG. 8, in these implementations, the pixelrange is calculated for windows 102, 104 of two different sizes. Thesize of the first window 102 is small (e.g., 3 pixels by 3 pixels) inorder to accurately estimate local contrast near edges. In someimplementations, the size of the second window 104 (e.g., 15 pixels by15 pixels) covers all of the shifted block transforms at a given pixel.The pixel range calculations result in a local contrast measure, lc, anda block contrast measure, bc.

The weights generator module 100 inputs the difference (Δ) between thelocal and block contrast measures (i.e., Δ(x,y)=bc(x,y)−lc(x,y)) into afunction α(Δ), which may be stored as a lookup table, to generate avalue of α(Δ) in the range [0,1]. Three different weighting functions106, 108, 110 are shown in FIG. 9. These functions correspond to thegeneral function given by equation (20):

$\begin{matrix}{{\alpha(\Delta)} = \frac{1}{1 + {\mathbb{e}}^{{- {({\Delta - T})}}/a}}} & (19)\end{matrix}$where a and T are variable parameters. A final lookup table for α(Δ),which was trained with test images using subjective evaluations andobjective PSNR (Peak Signal-To-Noise Ratio) values, is provided in Table2 below. This table is presented in “reading order” (i.e., left toright, top to bottom):

TABLE 2 Weights Lookup 0.000045, 0.000051, 0.000057, 0.000065, 0.000073,0.000082, 0.000092, 0.000103, 0.000116, 0.000130, 0.000147, 0.000165,0.000185, 0.000208, 0.000234, 0.000263, 0.000296, 0.000333, 0.000374,0.000421, 0.000473, 0.000532, 0.000598, 0.000672, 0.000755, 0.000849,0.000955, 0.001073, 0.001207, 0.001356, 0.001525, 0.001714, 0.001927,0.002166, 0.002434, 0.002736, 0.003075, 0.003456, 0.003884, 0.004365,0.004905, 0.005512, 0.006193, 0.006958, 0.007816, 0.008779, 0.009860,0.011072, 0.012432, 0.013956, 0.015664, 0.017577, 0.019719, 0.022117,0.024798, 0.027796, 0.031144, 0.034881, 0.039048, 0.043691, 0.048858,0.054601, 0.060975, 0.068040, 0.075858, 0.084493, 0.094010, 0.104477,0.115961, 0.128525, 0.142232, 0.157137, 0.173288, 0.190724, 0.209470,0.229535, 0.250913, 0.273574, 0.297470, 0.322526, 0.348645, 0.375706,0.403567, 0.432063, 0.461017, 0.490236, 0.519521, 0.548673, 0.577495,0.605799, 0.633410, 0.660172, 0.685949, 0.710628, 0.734120, 0.756358,0.777300, 0.796925, 0.815232, 0.832237, 0.847968, 0.862468, 0.875787,0.887984, 0.899121, 0.909265, 0.918480, 0.926835, 0.934395, 0.941223,0.947381, 0.952926, 0.957912, 0.962391, 0.966410, 0.970013, 0.973241,0.976129, 0.978713, 0.981022, 0.983085, 0.984927, 0.986572, 0.988039,0.989347, 0.990514, 0.991554, 0.992481, 0.993307, 0.994043, 0.994698,0.995282, 0.995801, 0.996264, 0.996676, 0.997042, 0.997368, 0.997659,0.997917, 0.998147, 0.998352, 0.998534, 0.998695, 0.998840, 0.998968,0.999082, 0.999183, 0.999274, 0.999354, 0.999425, 0.999489, 0.999545,0.999596, 0.999640, 0.999680, 0.999715, 0.999747, 0.999775, 0.999800,0.999822, 0.999842, 0.999859, 0.999875, 0.999889, 0.999901, 0.999912,0.999922, 0.999930, 0.999938, 0.999945, 0.999951, 0.999956, 0.999961,0.999965, 0.999969, 0.999973, 0.999976, 0.999978, 0.999981, 0.999983,0.999985, 0.999986, 0.999988, 0.999989, 0.999990, 0.999992, 0.999992,0.999993, 0.999994, 0.999995, 0.999995, 0.999996, 0.999996, 0.999997,0.999997, 0.999997, 0.999998, 0.999998, 0.999998, 0.999998, 0.999999,0.999999, 0.999999, 0.999999, 0.999999, 0.999999, 0.999999, 0.999999,0.999999, 0.999999, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000,1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000

Other embodiments are within the scope of the claims.

For example, although the above compression artifact reductionembodiments are described in connection with an image 12 that iscompressed by a block-transform-based image compression method, theseembodiments readily may be used to reduce artifacts in images compressedby other non-block-transform-based image compression techniques.

APPENDIX

The following computer code corresponds to an implementation of the fastfactorization shown in equations (6)-(9) for the forward denoisertransform of equation (4) and the corresponding factorization for thecorresponding inverse denoiser transform Z⁻¹. The inputs for thisimplementation are 64 element integer arrays.

void int_Forward_Denosier_Transform(int f[ ]) {  int c;  int * r, tmp0,tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;  int tmp10, tmp11, tmp12,tmp13;  for (r = f, c = 0; c < 8; c+ + , r+ +) {  // process columns  tmp0 = r[ 0] + r[56]; tmp7 = r[ 0] − r[56];   tmp1 = r[ 8] + r[48];tmp6 = r[ 8] − r[48];   tmp2 = r[16] + r[40]; tmp5 = r[16] − r[40];  tmp3 = r[24] + r[32]; tmp4 = r[24] − r[32];   tmp10 = tmp0 + tmp3;tmp13 = tmp0 − tmp3;   tmp11 = tmp1 + tmp2; tmp12 = tmp1 − tmp2;   r[0]= tmp10 + tmp11; r[32] = tmp10 − tmp11;   r[16] = tmp12 + (tmp13 < < 1);r[48] = tmp13 − (tmp12 < < 1);   /* Bottom half of the matrix */   tmp10= tmp4 + tmp7; tmp13 = tmp7 − tmp4;   tmp11 = tmp5 + tmp6; tmp12 = tmp5− tmp6;   r[40] = tmp10 − (tmp6 < < 1);   r[8] = tmp11 + (tmp7 < < 1);  r[56] = tmp12 − (tmp4 < < 1);   r[24] = tmp13 − (tmp5 < < 1);  }  for(r = f, c = 0; c < 8; c+ + , r + = 8) {  // process rows   tmp0 = r[0] +r[7]; tmp7 = r[0] − r[7];   tmp1 = r[1] + r[6]; tmp6 = r[1] − r[6];  tmp2 = r[2] + r[5]; tmp5 = r[2] − r[5];   tmp3 = r[3] + r[4]; tmp4 =r[3] − r[4];   tmp10 = tmp0 + tmp3; tmp13 = tmp0 − tmp3;   tmp11 =tmp1 + tmp2; tmp12 = tmp1 − tmp2;   r[0] = tmp10 + tmp11; r[4] = tmp10 −tmp11;   r[2] = tmp12 + (tmp13 < < 1); r[6] = tmp13 − (tmp12 < < 1);  /* Bottom half of the matrix */   tmp10 = tmp4 + tmp7; tmp13 = tmp7 −tmp4;   tmp11 = tmp5 + tmp6; tmp12 = tmp5 − tmp6;   r[5] = tmp10 − (tmp6< < 1);   r[1] = tmp11 + (tmp7 < < 1);   r[7] = tmp12 − (tmp4 < < 1);  r[3] = tmp13 − (tmp5 << 1);  } } voidint_Inverse_Denosier_Transform(int f[]) {  int c;  int * r, tmp0, tmp1,tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;  int tmp10, tmp11, tmp12, tmp13,z10,z11,z12,z13;  for (r = f, c = 0; c < 8; c+ +, r + = 8) { // processrows   tmp10 = r[0] + r[4]; tmp11 = r[0] − r[4];   tmp12 = r[2] − (r[6]< < 1); tmp13 = (r[2] < < 1) + r[6];   tmp0 = tmp10 + tmp13; tmp3 =tmp10 − tmp13;   tmp1 = tmp11 + tmp12; tmp2 = tmp11 − tmp12;   z10 =r[5] − r[3]; z13 = r[5] + r[3];   z11 = r[1] + r[7]; z12 = r[1] − r[7];  tmp4 = z10 − (r[7] < < 1);   tmp5 = z11 − (r[3] < < 1);   tmp6 = z12 −(r[5] < < 1);   tmp7 = z13 + (r[1] < < 1);   r[0] = tmp0 + tmp7; r[7] =tmp0 − tmp7;   r[1] = tmp1 + tmp6; r[6] = tmp1 − tmp6;   r[2] = tmp2 +tmp5; r[5] = tmp2 − tmp5;   r[3] = tmp3 + tmp4; r[4] = tmp3 − tmp4;  } for (r = f, c = 0; c < 8; c+ +, r+ +) { // process columns   tmp10 =r[0] + r[32]; tmp11 = r[0] − r[32];   tmp12 = r[16] − (r[48] < < 1);tmp13 = (r[16] < < 1) + r[48];   tmp0 = tmp10 + tmp13; tmp3 = tmp10 −tmp13;   tmp1 = tmp11 + tmp12; tmp2 = tmp11 − tmp12;   z10 = r[40] −r[24]; z13 = r[40] + r[24];   z11 = r[8] + r[56]; z12 = r[8] − r[56];  tmp4 = z10 − (r[56] < < 1);   tmp5 = z11 − (r[24] < < 1);   tmp6 = z12− (r[40] < < 1);   tmp7 = z13 + (r[8] < < 1);   r[0] = tmp0 + tmp7;r[56] = tmp0 − tmp7;   r[8] = tmp1 + tmp6; r[48] = tmp1 − tmp6;   r[16]= tmp2 + tmp5; r[40] = tmp2 − tmp5;   r[24] = tmp3 + tmp4; r[32] = tmp3− tmp4;  } }

1. A method of processing an input image compressed in accordance with ablock discrete cosine transform (DCT) image compression process,comprising: computing spatially-shifted forward transforms of the inputimage, each of the forward transforms being computed based on a denoisertransform Z having an associated transpose Z′, wherein a matrixmultiplication between Z and Z′ produces a diagonal matrix Λ, Z=F(D), Fspecifies a nonlinear mapping from coefficients of D to coefficients ofZ, and D substantially corresponds to a frequency-domain transform;denoising the forward transforms based on nonlinear mappings derivedfrom quantization values linked to the input image; computingspatially-shifted inverse transforms of the denoised forward transforms,each of the inverse transforms being computed based on Z and Z′; andcomputing an output image based on a combination of ones of thespatially-shifted inverse transforms.
 2. The method of claim 1, whereinD is a block-based linear transform.
 3. The method of claim 2, whereinthe computing of the spatially-shifted forward transforms comprisesapplying a forward transform operation to each of multiple positions ofa blocking grid relative to the input image and the spatially-shiftedinverse transforms are computed based on the relative positions of theblocking grid used to compute corresponding ones of thespatially-shifted forward transforms.
 4. The method of claim 2, whereinD is a discrete cosine transform.
 5. The method of claim 3, wherein D isa one-dimensional discrete cosine transform.
 6. The method of claim 5,wherein F is an arithmetic operator.
 7. The method of claim 6, wherein Fis a rounding arithmetic operator.
 8. The method of claim 1, wherein Fis a mapping from coefficients of D to corresponding coefficients of Zhaving values selected from 0 and ±2N where N has an integer value. 9.The method of claim 1, wherein F is a mathematical operatorcorresponding to one of a rounding operator, a floor operator, a ceilingoperator, and a truncate operator, and the computing of the forwardtransform coefficients comprises applying the mathematical operator tocoefficients of D weighted by respective scaling factors to obtaincorresponding coefficients of Z.
 10. The method of claim 9, wherein thecomputing of the forward transform coefficients comprises weighting thecoefficient of D by a common scaling factor.
 11. The method of claim 10,wherein F corresponds to a rounding operator applied to the weightedcoefficients of D.
 12. The method of claim 9, wherein the denoisingcomprises applying to the forward transforms nonlinear mappings derivedfrom the quantization values linked to the input image and weighted byrespective scaling factors.
 13. The method of claim 1, wherein thecomputing of the forward transforms is based on a factorization of Z.14. The method of claim 1, wherein the input image corresponds to adecompressed version of an input image compressed based on a givenquantization process and the forward transforms are denoised based onthe given quantization process.
 15. The method of claim 1, wherein theforward transforms are denoised by setting to zero each of the forwardtransform coefficients with an absolute value below a respectivethreshold derived from a respective one of the quantization valueslinked to the input image and leaving unchanged each of the forwardtransform coefficients with an absolute equal to at least a respectivethreshold derived from a respective ones of the quantization valueslinked to the input image.
 16. The method of claim 15, furthercomprising sharpening the forward transform coefficients by increasingnonlinear transform parameters by respective factors that are larger forhigher spatial frequency forward transform coefficients than for lowerspatial frequency forward transform coefficients.
 17. The method ofclaim 1, wherein the output image is computed from a weightedcombination of the inverse transforms.
 18. The method of claim 17,wherein the computed output image corresponds to an average of theinverse transforms.
 19. The method of claim 1, whereinZ_(ij)=round{(3.5)·D_(ij)}, round{} is a rounding operator, Z_(ij), is acoefficient of Z in row i and column j, and D_(ij) is the coefficient ofa discrete cosine transform in row i and column j.
 20. A method ofprocessing an input image, comprising: computing spatially-shiftedforward transforms of the input image, each of the forward transformsbeing computed based on a denoiser transform Z having an associatedtranspose Z′, wherein a matrix multiplication between Z and Z′ producesa diagonal matrix Λ, Z=F(D), F specifies a mapping from coefficients ofD weighted by a common scaling factor to corresponding coefficients ofZ, and D substantially corresponds to a frequency-domain transform;denoising the forward transforms based on nonlinear mappings derivedfrom quantization values linked to the input image, wherein thenonlinear mappings are derived from quantization values weighted by thecommon scaling factor; computing spatially-shifted inverse transforms ofthe denoised forward transforms, each of the inverse transforms beingcomputed based on Z and Z′; and computing an output image based on acombination of ones of the spatially-shifted inverse transforms.
 21. Amethod of processing an input image, comprising: computingspatially-shifted forward transforms of the input image, each of theforward transforms being computed based on a denoiser transform Z havingan associated transpose Z′, wherein a matrix multiplication between Zand Z′ produces a diagonal matrix Λ, Z=F(D), F specifies a nonlinearmapping from coefficients of D to coefficients of Z, and D substantiallycorresponds to a frequency-domain transform; denoising the forwardtransforms based on nonlinear mappings derived from quantization valueslinked to the input image; computing spatially-shifted inversetransforms of the denoised forward transforms, each of the inversetransforms being computed based on Z and Z′; and computing an outputimage based on a combination of ones of the spatially-shifted inversetransforms, wherein computing the output image comprises computing abase image from a combination of ones of the inverse transforms,computing a ringing correction image based at least in part on computedmeasures of local spatial intensity variability for pixels of each ofthe inverse transforms, and combining pixel values from the base imageand the ringing correction image.
 22. The method of claim 21, whereinthe base image has pixel values corresponding to respective averages ofvalues of corresponding pixels in the inverse transforms.
 23. The methodof claim 21, further comprising assigning to each pixel in the ringingcorrection image a value of a corresponding inverse transform pixelhaving a lowest computed measure of local spatial intensity variabilityof the corresponding inverse transform pixels.
 24. The method of claim21, further comprising assigning to each pixel in the ringing correctionimage a value corresponding to an average of multiple correspondinginverse transform pixels in a lowest percentile of local spatialvariability measures of the corresponding inverse transform pixels. 25.The method of claim 21, wherein the output image is computed by aweighted combination of the base image and the ringing correction image.26. The method of claim 21, wherein in the combining the base imagecontributes less to the output image than the ringing correction imagefor pixels adjacent to transition regions in the base image.
 27. Asystem for processing an input image compressed in accordance with ablock discrete cosine transform (DCT) image compression process,comprising: a forward transform module configured to computespatially-shifted forward transforms of the input image, each of theforward transforms being computed based on a denoiser transform Z havingan associated transpose Z′, wherein a matrix multiplication between Zand Z′ produces a diagonal matrix Λ, Z=F(D), F specifies a nonlinearmapping from coefficients of D to coefficients of Z, and D substantiallycorresponds to a frequency-domain transform; a nonlinear denoiser moduleconfigured to denoise the forward transforms based on nonlinear mappingsderived from quantization values linked to the input image; an inversetransform module configured to compute spatially-shifted inversetransforms of the denoised forward transforms based on Z and Z′; and anoutput image generator module configured to compute an output imagebased on a combination of ones of the spatially-shifted inversetransforms.
 28. A computer-readable medium storing computer-readableinstructions for processing an image compressed in accordance with ablock discrete cosine transform (DCT) image compression process, thecomputer-readable instructions being executable to cause a computer toperform operations comprising: computing spatially-shifted forwardtransforms of the input image, each of the forward transforms beingcomputed based on a denoiser transform Z having an associated transposeZ′, wherein a matrix multiplication between Z and Z′ produces a diagonalmatrix Λ, Z=F(D), F specifies a nonlinear mapping from coefficients of Dto coefficients of Z, and D substantially corresponds to afrequency-domain transform; denoising the forward transforms based onnonlinear mappings derived from quantization values linked to the inputimage; computing spatially-shifted inverse transforms of the denoisedforward transforms based on Z and Z′; and computing an output imagebased on a combination of ones of the spatially-shifted inversetransforms.